Projects per year

### Abstract

We give an algorithm for computing all roots of polynomials over a univariate power series ring over an exact field K. More precisely, given a precision d, and a polynomial Q whose coefficients are power series in x, the algorithm computes a representation of all power series f(x) such that Q(f(x)) = 0 mod x^{d}. The algorithm works unconditionally, in particular also with multiple roots, where Newton iteration fails. Our main motivation comes from coding theory where instances of this problem arise and multiple roots must be handled. The cost bound for our algorithm matches the worst-case input and output size d deg(Q), up to logarithmic factors. This improves upon previous algorithms which were quadratic in at least one of d and deg(Q). Our algorithm is a refinement of a divide & conquer algorithm by Alekhnovich (2005), where the cost of recursive steps is better controlled via the computation of a factor of Q which has a smaller degree while preserving the roots.

Original language | English |
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Title of host publication | ISSAC 2017 - Proceedings of the 2017 ACM International Symposium on Symbolic and Algebraic Computation |

Number of pages | 8 |

Volume | Part F129312 |

Publisher | Association for Computing Machinery |

Publication date | 23 Jul 2017 |

Pages | 349-356 |

ISBN (Electronic) | 9781450350648 |

DOIs | |

Publication status | Published - 23 Jul 2017 |

Event | 42nd ACM International Symposium on Symbolic and Algebraic Computation, ISSAC 2017 - University of Kaiserslautern, Kaiserslautern, Germany Duration: 25 Jul 2017 → 28 Jul 2017 Conference number: 42 |

### Conference

Conference | 42nd ACM International Symposium on Symbolic and Algebraic Computation, ISSAC 2017 |
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Number | 42 |

Location | University of Kaiserslautern |

Country | Germany |

City | Kaiserslautern |

Period | 25/07/2017 → 28/07/2017 |

Sponsor | Association for Computing Machinery |

### Keywords

- List decoding
- Polynomial root-finding algorithm
- Power series

## Projects

- 1 Finished

## COFUNDPostdocDTU: COFUNDPostdocDTU

Præstrud, M. R. & Brodersen, S. W.

01/01/2014 → 31/12/2019

Project: Research

## Cite this

*ISSAC 2017 - Proceedings of the 2017 ACM International Symposium on Symbolic and Algebraic Computation*(Vol. Part F129312, pp. 349-356). Association for Computing Machinery. https://doi.org/10.1145/3087604.3087642